In mathematics, substitution is a technique used to simplify an equation or expression by replacing a variable with another expression or value. Substitution is commonly used in algebra, calculus, and differential equations.

Here are a few contexts where substitution is applied:

### 1. **Algebraic Substitution**:
   You replace a variable with a number or another expression. For example, in the equation \( y = 2x + 3 \), if you know \( x = 4 \), you substitute \( 4 \) for \( x \), which gives \( y = 2(4) + 3 = 11 \).

### 2. **Integration by Substitution** (Calculus):
   This technique simplifies an integral by making a substitution, typically with \( u \)-substitution. For instance:
   \[
   \int (2x) \, dx
   \]
   By substituting \( u = 2x \), you can rewrite the integral in terms of \( u \).

### 3. **Trigonometric Substitution**:
   Used to simplify integrals involving square roots. For example, if an integral has \( \sqrt{a^2 - x^2} \), you can use the substitution \( x = a \sin(\theta) \) to simplify it.

### 4. **Solving Systems of Equations**:
   Substitution is often used to solve systems of linear equations. For example, in the system:
   \[
   y = 2x + 3 \quad \text{and} \quad x + y = 7
   \]
   You can substitute \( y = 2x + 3 \) into the second equation to find \( x \).

### 5. **Differential Equations**:
   In some cases, substitution simplifies a differential equation. For instance, substituting a new variable might reduce a nonlinear equation to a linear one.

Would you like more details on any specific substitution method?

Here are some related questions to deepen understanding:
1. How do you solve systems of equations using substitution?
2. Can you explain how \( u \)-substitution works in calculus with an example?
3. How is trigonometric substitution applied in integrals involving square roots?
4. What are the advantages of substitution compared to other methods like elimination in solving equations?
5. How is substitution used in solving first-order differential equations?

**Tip**: When using substitution in integrals, always remember to change both the limits of integration (if definite) and the differential term to avoid mistakes!

Explore the technique of substitution in mathematics, commonly used in algebra, calculus, and solving differential equations. This comprehensive guide covers algebraic substitution, u-substitution in integrals, trigonometric substitution, and solving systems of equations. Suitable for students in Grades 10-12.

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